I have watched the Gilber Strang's great course. Lec 22 | MIT 18.06 Linear Algebra, explains everything but what the eigenvectors are. After some struggling I have figured out that companion matrix turns eighenvector [a b c d] into [λa λb λc λd]. Herein, λ is vector's the eigenvalue. Now, what are the [a, b, c, etc]? They are [λ³ λ² λ 1] because

*a*element of the input vector turns into*λa*at the next iteration! Next time,*λa*turns into*λ**²a.*This way, you must conclude that the train of values is the vector [λ³ λ² λ 1]. If we recall that it corresponds to [y₃ y₂ y₁ y₀], we'll understand that we start in y₀, next state variable y₁ = λy₀, y₂ = λ²y₀ and progress so on. But, we can factor the y₀ out of the eigenvector, [y₃ y₂ y₁ y₀] = y₀ [λ³ λ² λ 1] and, because all eigenvectors are identical up the scale factor, we can safely eliminate the y₀, yielding the eigenvector of [λ³ λ² λ 1]. It is perfect that this vector is a geometric series (aka exponential function), which is an eigenfunction of any differential and difference equation.
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